Algebraic equations can be used to interpret situations and find answers to questions in a wide range of contexts. The key to solving these problems is in constructing the algebraic expressions needed to represent the important details.

Examples

Example 1

Adam sells chocolates to raise money for charity. Each chocolate costs \$6.

If Adam sells q chocolates, write an algebraic expression for the amount of money he raises.

Worked Solution

Create a strategy

To find the amount of money Adam raises, we can use multiplication.

Apply the idea

So the algebraic expression for the amount of money Adam raises is 6q.

Which statement is correct?

It is possible to raise \$13 if adam sells chocolates at \$6.

4 chocolates at \$6 each will cost \$10.

It is possible for Adam to raise only \$1 after sales.

5 chocolates at \$6 each will cost \$30.

Worked Solution

Create a strategy

Use the expression from part (a) and check each statement.

Apply the idea

6q is the amount that Adam raises when he sells q chocolates.

Adam's sales should always be a multiple of \$6, the cost of each chocolate.

So options A, B and C are not correct since the amounts are not multiples of \$6.

Let's check option D:

\displaystyle 6q	\displaystyle =	\displaystyle 6\times 5	Replace q with 5
	\displaystyle =	\displaystyle \$30	Evaluate the product

So, the correct statement is option D: 5 chocolates at \$6 each will cost \$30.

Example 2

Judy has 72 pencils, which she shares evenly among the students in her class.

If there are q students in her class, write an algebraic expression for the number of pencils given to each student.

Worked Solution

Create a strategy

To find the expression we can then divide the number of pencils by the number of students.

Apply the idea

Judy is sharing 72 pencils and among q students.

So the algebraic expression for the number of pencils given to each student is \dfrac{72}{q}.

Which statement is correct?

The more students that are in the class, the less pencils each student receives.

The more students that are in the class, the more pencils Judy gives out.

If there were 2 more students in the class, each student would recieve 2 less pencils.

The less students that are in the class, the less pencils each student receives.

Worked Solution

Create a strategy

Use the expression from part (a) and substitute the value.

Apply the idea

We want to know what happens when the number of students increases or decreases by substituting the value of q into the equation.

If the pencils are shared among 2 students, each would get:

\displaystyle \frac{72}{q}	\displaystyle =	\displaystyle \frac{72}{2}	Substitute q=2
	\displaystyle =	\displaystyle 32 \text{ pencils}

If the pencils are shared among 3 students, each would get:

\displaystyle \frac{72}{q}	\displaystyle =	\displaystyle \frac{72}{3}	Substitute q=3
	\displaystyle =	\displaystyle 24 \text{ pencils}

The more students that are in the class, the less pencils each student receives. So the correct statement is in option A.

Idea summary

The key to solving contextual problems is in constructing the algebraic expressions needed to represent the important details. Then we can use substitution to answer the question.

Outcomes

ACMNA176

Create algebraic expressions and evaluate them by substituting a given value for each variable

ACMNA177

Extend and apply the laws and properties of arithmetic to algebraic terms and expressions

6.06 Solving contextual problems with algebra

Ideas

Contextual problems with algebra

Examples

Example 1

Example 2

Outcomes

ACMNA176

ACMNA177

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